Integral representation of vertical operators on the Bergman space over the upper half-plane by Bais, Shubham R., Venku Naidu, D., Mohan, Pinlodi

“…In this article, we prove that every vertical operator on the Bergman space $\mathcal{A}^2(\Pi )$ over the upper half-plane can be uniquely represented as an integral operator of the form \begin{equation*} \left(S_\varphi f\right)(z) = \int _{\Pi } f(w) \varphi (z-\overline{w}) d\mu (w),~~\forall f\in \mathcal{A}^2(\Pi ),~z\in \Pi , \end{equation*} where $\varphi $ is an analytic function on $\Pi $ given by \begin{equation*} \varphi (z) = \int _{\mathbb{R}_+}\xi \sigma (\xi ) e^{iz\xi } d\xi , \ \forall z\in \Pi \end{equation*} for some $\sigma \in L^\infty (\mathbb{R}_+)$. …”
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